Modelling wealth with a Pareto distribution: how do I estimate the parameters?I wish to create a function that will estimate the wealth of a person in the United States. It would be used to make a table with each decile and their estimated wealth.This estimate will be based on very rudimentary data, and is only for personal interest. The data is:- The total wealth of the bottom 90% is equal to the total wealth of the top 0.1%.- Both proportions have 22% of the total wealth.- The total wealth under the distribution is $80 trillion.- The total population is 160 million households.Given this data, how would I create parameter estimates for the exponent and scale of a pareto distribution? What would be f(x) where x is from (0,1), and the solution is the wealth of someone richer than that proportion of people? For example f(0.1) is someone richer than or equal to exactly 10% of the least wealthy, and could equal 1,000 dollars. F(0.5) is the median wealth, and could be 200,000 dollars. F(0.9999) is richer than 99.99% and would be somewhere in the tens or hundreds of millions of dollars.

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RamPatWeese2w · Accepted Answer

Step 1First of all, the Pareto distribution is defined on non-negative real numbers. Thus it doesn&#039;t really make sense to force it to the interval (0,1).Suppose, however, you knew the wealth of two people: someone at the 90th percentile and the 99.90th percentile. Then, if you assume that wealth is truly Pareto-distributed, you could take a method-of-moments approach.Letting A be the wealth of the 90th percentile person, and B the wealth of the 99.90th percentile person, and F(x) be the cumulative distribution function of a Pareto(      x    m    ,  α) random variable, we have that  F  (  x  )  =  1  −  (            x      m        x        )    α      ⟺        F          −      1        (  y  )  =            x      m              (      1      −      y              )                  1                      /                    α                        ⟺        x    m    =  x  (  1  −  y      )          1              /            α      Step 2From two statements that you provided, we have that the person with wealth B has greater wealth than 78% of the population, and that the person with wealth A has greater wealth than 22% of the population. You could then estimate the Pareto distribution parameters by solving the equations      x    m    =  A  ⋅  (  1  −  0.22      )          1              /            α            x    m    =  B  ⋅  (  1  −  0.78      )          1              /            α

Modelling wealth with a Pareto distribution: how do I estimate the parameters? I wish to create a function that will estimate the wealth of a person in the United States. It would be used to make a table with each decile and their estimated wealth.

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